Questions tagged [line-bundles]
A continuously varying family of one-dimensional vector spaces over a topological space. A related tag is the vector-bundles tag.
190
questions
4
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127 views
Non-trivial line bundle on $\mathbb{C}^{\ast} \times \mathbb{C}^{\ast}$
A line bundle is a holomorphic complex-dimension-one bundle on a complex manifold.
The complex manifold $X = \mathbb{C}^{\ast} \times \mathbb{C}^{\ast}$ admits a non-trivial line bundle for the ...
2
votes
1answer
47 views
Grassmannian of line subbundle of a stable rank 2 vector bundle on a smooth projective curve
Let $X$ be a smooth projective curve of genus $g\geq 2$. Given a rank two, degree $d=0$ vector bundle $\mathcal{F}$ on $X$, we consider the grassmannian of sub-line bundles of $\mathcal{F}$ of degree $...
1
vote
1answer
131 views
Riemannian vector bundle [closed]
I'm trying to show the curvature of a one dimensional vector bundle with a Riemannian metric vanishes, no matter what the connection is. I found this can be done for orientable bundles, because an ...
2
votes
1answer
190 views
Galois invariant line bundle and base change
Let $K$ be a number field and consider a finite Galois extension $L|K$. Moreover let $X$ be a projective, regular, integral variety over $K$. After a base change we obtain a morphism of varieties $f:...
1
vote
0answers
118 views
Restriction of a line bundle on $G/B$ to a fibre which is isomorphic to $\mathbb{P}^1$
Let $G$ be a reductive group over a field $k$ of characteristic zero with maximal split torus $T$, Borel $B \supset T$ and Weyl group $W$. Set $X:=G/B$ and $C_w:=BwB/B \subset X$ for $w \in W$ the ...
3
votes
2answers
124 views
Character which defines canonical bundle on flag variety
Let $G$ be a reductive group over a field $k$ of characteristic zero with maximal split torus $T$ and Borel $B \supset T$ defining a set of simple roots $\Delta$. Additionally let $\rho$ be the half ...
1
vote
0answers
82 views
From a factor of automorphy on an abelian variety to a divisor
Given a complex abelian variety $A = V/\Gamma$ (for $\Gamma$ being a lattice in the complex vector space $V$), one knows how to describe a holomorphic line bundle in terms of factors of automorphy: By ...
1
vote
1answer
241 views
Making a vector bundle ample by twisting with ample line bundle
Let $X$ be a projective algebraic variety over some field (I am happy to add some more assumptions if necessary). A vector bundle $E$ is ample if the relative twisting sheaf $\mathcal{O}_{\mathbf{P}(E)...
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0answers
211 views
Construction of the Hilbert Scheme
I am reading the book "Rational Curves on Algebraic Varieties" of János Kollár. Definition-Proposition 1.2, begin like this:
Let $g:Y\rightarrow Z$ be a projective morphism and $\mathcal{O}(...
0
votes
0answers
130 views
dimension of linear system and multiplicity at a point
I recently encountered the following statement which I am unable to prove.
Let $X$ be a smooth projective surface and let $L$ be a line bundle on $X$. For $x\in X$ if $h^0(|L|)\geq\frac{m(m+1)}{2}$ ...
0
votes
1answer
206 views
Holomorphic sections to anti-holomorphic sections
Let $X$ be a compact Kähler manifold and $L$ be a holomorphic line bundle on $X$ with a Hermitian metric $h$. I am trying to give a norm preserving isomorphism between the space of holomorphic ...
4
votes
0answers
75 views
Classification of square roots of line bundles and metalinear/metaplectic structures
Reading some books and articles about geometric quantization I got confused about the classification of square roots of complex line bundles over a manifold. Consider the group of isomorphism classes ...
1
vote
1answer
129 views
Flat connection of a degree zero line bundle on curve
The question is clear from the title. Suppose we have a line bundle on a compact smooth complex curve $X$, and a line bundle $\mathcal{L}=\mathcal{O}_X(p-q)$, where $p$ and $q$ are divisors, then what ...
1
vote
1answer
230 views
(Bridgeland stability conditions)How to get the heart of a bounded t-structure on $D^b(P^1 \times P^2)$?
I have already known how to get the heart of a bounded t-structure on $D^b(P^n)$ by Macri`s paper,
https://arxiv.org/abs/math/0411613.
However I cannot purpose analogously on $D^b(P^1 \times P^2)$.
...
4
votes
0answers
55 views
Isometries of fiber bundles
Let $F\to S\overset{\pi}{\to} B$ a Riemannian submersion with totally geodesic fibers.
Question: How much information about the isometries of $S$ we have if we know the isometries of $F$ and $B$? For ...
1
vote
0answers
152 views
Proof of uniqueness in the universal property of Poincaré line bundles
My question concerns the proof of a part of Lemma IV.2.2 (pag. 168) of the book Geometry of Algebraic Curves. vol. I by Arbarello, Cornalba, Griffiths and Harris. In order to state my problem, let me ...
4
votes
1answer
106 views
Identity relating iterated determinant line bundles
Suppose that $R$ is a (commutative, unital) ring and that $A$ is a (commutative, unital) $R$-algebra that is projective of constant rank $n$ as an $R$-module. Then $A$ has a "determinant line ...
0
votes
0answers
113 views
Sections of vector bundles interpreted as sections of line bundles
Let $X$ be a smooth projective curve of genus $g$ over $\mathbb{C}$, $K_{X}$ be a cononical sheaf on $X$ and $\mathcal{E}$ be a locally free sheaf on $X$ s.t. $H^{0}(X,\mathcal{E}^{*})=\operatorname{...
0
votes
0answers
174 views
Dimension of global holomorphic sections of a line bundle
Let $K$ be the canonical line bundle of a compact Riemann surface $M$ of genus $g$. Consider the pull back of $K$ on $M \times M$ via projection on the first factor. What is the dimension of the space ...
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0answers
78 views
Discriminant divisor $\mathcal{D}_{r} \subseteq H^{0}(X,K_{X}^{\otimes r})$ is irreducible
Let $X\colon$ smooth projective curve over $\mathbb{C}$, $K_{X}\colon$ canonical line bundle over $X$, and $W_{r}$ denotes $H^{0}(X,K_{X}^{\otimes r})$.
I'm trying to prove the following proposition,
...
2
votes
0answers
142 views
What is $f^*TX$ for a general morphism $f\colon\mathbb{P}^1\to X$?
Let $X$ be a projective homogeneous space over $\mathbb{C}$, i.e. $G/P$ where $G$ is a simple, simply connected linear algebraic group and $P$ is a parabolic subgroup. Let $f\colon\mathbb{P}^1\to X$ ...
9
votes
2answers
279 views
Differential refinement of homology
Differential cohomology is a refinement of ordinary cohomology by differential data. It's construction comes down to the observation that $H^2(M, \mathbb{Z})$ is isomorphic to the space of isomorphism ...
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0answers
89 views
Vector bundle defined by using divisors of very ample line bundle
Let $X$ be a smooth projective curve. Suppose that $L_1$ and $L_2$ are line bundles on $X$, and $L_1$ is very ample.
$\operatorname{Div}(s)$ denotes a divisor defined by a global section $s\in H^0(X,L)...
7
votes
1answer
218 views
The set of isomorphism classes of Z/nZ-equivariant line bundles over a 2 dimensional Z/nZ-CW complex
Suppose I wish to find the set of isomorphism classes of $\mathbb{Z}/n\mathbb{Z}$-equivariant line bundles over a 2-dimensional, compact $\mathbb{Z}/n\mathbb{Z}$-CW-complex $X$, i.e. $\mathrm{Vect}^{1}...
1
vote
0answers
210 views
Embeddings of Hirzebruch surfaces $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(n))$
Let $X_n=\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(n))$ be the $n-$th Hirzebruch surface. We know that for $d>0$ and higher $k>>0$ the linear system $$\mathcal{L}_{...
7
votes
1answer
228 views
Multiplication maps for big line bundles
In Birational Geometry of Algebraic Varieties, Kollar and Mori write that for a line bundle "being big is essentially the birational version of being ample" (page 67). Recall that a line ...
5
votes
1answer
370 views
Line bundle on product scheme
Let $k$ be a field, $X$ be a complete variety over $k$, $V$ be an open subvariety of $X$, $Y$ be a scheme over $k$. Suppose $L$ is a line bundle on $V\times Y$. If $L|_{V\times\lbrace y\rbrace}$ ...
3
votes
1answer
110 views
Weights on the linearization
Consider, just as an example, an action of $\mathbb{C}^*$ on $\mathbb{P}^2$ of the form
$$t\cdot p=[p_0:tp_1:t^2p_2]$$
There are $3$ fixed points, namely $e_1,e_2,e_3$. If I consider a $\mathbb{C}^*$-...
3
votes
1answer
308 views
Weak Lefschetz theorem for Lef line bundles
I'm studying
M. A. A. de Cataldo, L. Migliorini - The Hard Lefschetz Theorem and the topology of semismall maps, Ann. sci. École Norm. Sup., Serie 4 35 (2002) 759-772.
The premises are the following....
1
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0answers
118 views
Problem regarding existence of a divisor representing line bundle
We consider a normal irreducible variety $X$ and a line bundle $L$. The question is when $L$ is induced by a Cartier divisor $D$. We know that if $s$ is a rational section of $O_X(D)$, where $D$ is a ...
1
vote
0answers
126 views
The group of global sections of the automorphism bundle of the tangent bundle on a Grassmannian
Let $X={\rm Gr}(k,n)$ denote the Grassmannian of $k$-dimensional subspaces in ${\Bbb C}^n$.
We regard $X$ as an algebraic variety over $\Bbb C$.
Let ${T_X} \to X$ denote the tangent bundle on $X$. For ...
2
votes
1answer
213 views
Polarization of an abelian variety made by the sum of two divisors
Let $X$ be an abelian variety of dimension $n$, and let $L$ be a polarization, that is, an ample line bundle on $X$, with $\chi(L)=3$.
In my specific case, I have that $L=\mathcal{O}_X(\Theta + D)$, ...
1
vote
0answers
232 views
Splitting principle in algebraic geometry and ample line bundles
Splitting theorem in algebraic geometry claims that if we have a vector bundle $V$ on $X$ (we consider a smooth projective variety for this question), if we pull-back $V$ to $\mathbb{P}(V)$, we get a ...
1
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0answers
363 views
Questions on Néron–Severi group
$\DeclareMathOperator\NS{NS}\DeclareMathOperator\Pic{Pic}$I have two questions on a comment from Daniel Hyubrechts's Complex Geometry on pages 133/134.
Let $X$ be a compact Kähler manifold. Consider ...
1
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0answers
135 views
Ample line bundle gives alternative description of a variety
Let $X$ be a (smooth) projective variety (over $\mathbb{C}$), and $\mathcal{L}$ an ample line bundle on $X$. I have heard that then
$$ X \cong \mathrm{Proj} \left( \bigoplus_{k \ge 0} H^0(X,\mathcal{...
3
votes
0answers
127 views
A question on Okounkov bodies
Let $X$ be an irreducible $n$-dimensional projective variety, and
$$Y_n\subset Y_{n-1}\subset\dots\subset Y_1\subset X$$
a flag of irreducible subvarieties such that $Y_i$ has codimension $i$ in $X$ ...
3
votes
0answers
166 views
Is Wronskian a line bundle for Riemann surfaces?
Suppose $f_1,\dots,f_g$ are holomorphic functons on a domain $U\subset\mathbb{C}$. By the Wronskian determinant $f_1,\dots,f_g$ one means the determinant of the matrix of derivatives $f_k^{(m)},$ ...
6
votes
0answers
120 views
Does there exist a notion of Chern classes in intersection cohomology?
First of all: I apologize for my mistakes, I'm a freshman in intersection cohomology.
Let $X$ be a (compact) complex analytic space, let $L$ be a line bundle over $X$.
Can one define a notion of ...
1
vote
1answer
129 views
Generators of a graded algebra defining bundle over elliptic curve
I have a question about a statement from P. Wagreich's paper "Elliptic Singularities of Surfaces" (page 425):
We consider an elliptic curve $X$ and a line bundle (=invertible sheaf) $L$ on $X$.
Then,...
3
votes
1answer
134 views
Sections of Cartier divisors on toric varieties
Let $X_{\Sigma}$ be a projective toric variety. Consider the total coordinate ring
$$S = \mathbb{C}[x_{\rho}\: | \: \rho\in\Sigma(1)]$$
Define $\deg(x_{\rho}) = D_{\rho}$.
Now, take a divisor $D = \...
1
vote
0answers
187 views
Are torsion-free rank 1 modules over integral schemes line bundles?
How far away are torsion-free rank 1 sheaves from the line bundles? Is there any condition that makes sure they are same? (for dimensions higher than 1). It is known that for a regular scheme of ...
3
votes
2answers
266 views
Line bundles trivial outside of codimension 3
Let $X$ be a CW complex (possibly a topological/smooth manifold) of dimension $n$, $L\to X$ a complex line bundle and $Y\subset X$ a subcomplex (possibly a submanifold) contained in the codimension 3 ...
3
votes
2answers
212 views
Existence of a connection $A$ on a holomorphic line bundle $L$, s.t $F(A)=(\deg L)\omega$
I'm reading this paper and at page 67, he states that for any line bundle $L$ over a Rieman surface there is a connection $A$ whose curvature is
$$
F(A)=(\deg L)\omega,
$$
where $\omega$ is a positive ...
5
votes
0answers
147 views
Arnold's theorem on small denominators and holomorphic tubular neighborhoods
By a theorem of Grauert, along a curve with negative self-intersection a complex surface is locally biholomorphic to a neighborhood of the zero section of that curve inside its normal bundle. For ...
2
votes
0answers
88 views
The dual of the space of continuous sections in a vector bundle
If $X$ is a compact Hausdorff space, one may view the space of complex, continuous functions on it as the space of continuous sections in the trivial Hermitian bundle $X \times \mathbb C$. By the ...
18
votes
1answer
995 views
Embedding abelian varieties into projective spaces of small dimension
Given a (complex) abelian variety $A$ of a fixed dimension $g$, let $d(A)$ be the dimension of the smallest complex projective space it embeds into.
Is $d(A)$ uniform over all abelian varieties of a ...
5
votes
1answer
196 views
The existence of the extension of a non-trivial line bundle
In Three Dimensional Gravity Revisted, Witten studied the Abelian Chern-Simons theory in three dimensions.
Let $W$ be a three dimensional manifold. Let $\mathcal{L}$ be a non-trivial line-bundle over ...
2
votes
1answer
116 views
Pull-back of polarization
Let $(X, L)$ and $(Y, M)$ be two polarized abelian varieties .
According to Birkenhake C. and Lange H. in Complex Abelian Varieties a homomorphism of polarized abelian varieties $f:(Y, M)\...
4
votes
0answers
166 views
Does there exist a preferred trivialization of a trivial line bundle?
Let $L\to M$ be a topologically trivial complex Hermitian line bundle (over a manifold of dimension three, if this is of any importance). I assume that $L$ admits a trivialization, however, I do not ...
2
votes
3answers
1k views
Classification of line bundles by second cohomology of a manifold
In the book Loop spaces, Characteristic classes and geometric quantization by Brylinski I see following result when trying to motivate geometric description of $H^3(M,\mathbb{Z})$.
$H^2(M,\mathbb{Z}...
